Theorem nfim1 1811

Description: A closed form of nfim 1813. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

Ref	Expression
nfim1.1	⊢ Ⅎxφ
nfim1.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfim1	⊢ Ⅎx(φ → ψ)

Proof of Theorem nfim1

Step	Hyp	Ref	Expression
1		nfim1.1	. . . 4 ⊢ Ⅎxφ
2	1	nfri 1762	. . 3 ⊢ (φ → ∀xφ)
3		nfim1.2	. . . 4 ⊢ (φ → Ⅎxψ)
4	3	nfrd 1763	. . 3 ⊢ (φ → (ψ → ∀xψ))
5	2, 4	hbim1 1810	. 2 ⊢ ((φ → ψ) → ∀x(φ → ψ))
6	5	nfi 1551	1 ⊢ Ⅎx(φ → ψ)

Syntax hints:   → wi 4  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfim  1813  sbco2d  2087